How to animate rotation of an object without modifiers
Parent your object to an empty and rotate the empty. So, in details:


Create your object, put its origin to its geometry, create 2 empties at the exact same point, parent the object to one of the empties, let say the cube.





Give your object a first Simple Deform modifier, Twist mode, Angle 360°.





Give your object a second Simple Deform modifier, Bend mode, Angle 360°, with the other empty (here a Plain Axes) as Axis. Now when you rotate the first empty (Cube) on the Z axis it should rotate the way you want.